Decomposable (4, 7) solutions in eleven-dimensional supergravity

Abstract

We describe a class of decomposable eleven-dimensional supergravity backgrounds (M 10,1 = M∼ 3,1 × M 7 , gM = g∼ + g) which are products of a four-dimensional Lorentzian manifold and a seven-dimensional Riemannian manifold, endowed with a flux form given in terms of the volume form on M∼ 3,1 and a closed 4-form F 4 on M 7 . We show that the Maxwell equation for such a flux form can be read in terms of the co-closed 3-form ψ = ∗ 7 F 4 . Moreover, the supergravity equation reduces to the condition that (M∼ 3,1 , g∼) is an Einstein manifold with negative Einstein constant and (M7,g,F) is a Riemannian manifold which satisfies the Einstein equation with a stressenergy tensor associated to the 3-form ψ. Whenever this 3-form is generic, we show that the Maxwell equation induces a weak G2-structure on M 7 and obtain decomposable supergravity backgrounds given by the product of a weak G2-manifold (M 7 , ψ, g) with a Lorentzian Einstein manifold (M∼ 3,1 , g∼). We also construct examples of compact homogeneous Riemannian 7-manifolds endowed with non-generic invariant 3-forms which satisfy the Maxwell equation, but the construction of decomposable homogeneous supergravity backgrounds of this type remains an open problem. © 2019 IOP Publishing Ltd.